Let: $$ f(x)=\frac{1-e^x}{x} $$ Then: $$ \lim_{x \to 0}f(x) = -1 $$ Question: I am looking for the way to express the rate of convergence of this limit in relation to a polynomial, using the big-Oh notation.
Attempt: a polynomial function that has the same limit is: $$ g(x)= -1-x $$ And by looking at the graphs of $f(x)$ and $g(x)$, we can see that: $$ |f(x)|<|g(x)| \text{, for small x} $$ So I believe I can write: $$ f(x) = -1 + O(x) \text{, as x $\to 0 $} $$ Please let me know if my thought was correct. If yes, how do I formally derive the result without relying on the graph? Otherwise, please give me some hints to solve this problem.